We have that the strong convexity constant using [1]} is \(\rho = \frac{1}{k}\)  w.r.t the \(\left\Vert .\right\Vert _1\)  over \(\mathcal {X}\) . The dual norm of \(\left\Vert .\right\Vert _1\)  is \(\left\Vert .\right\Vert _\infty \) , take \(p \rightarrow \infty \)  in (REF ) to get \(L = \left\Vert \mathbf {w}\right\Vert _\infty h\) . We use the general regret bound in [2]} for OMD strategies, to obtain:
\(\mathrm {Regret}_T(\mathrm {OMD}_\Phi ) \le \frac{D_{\Phi }(\mathbf {x}_*, \mathbf {x}_1)}{\eta } + \frac{\eta }{2 \rho } \sum ^T_{t=1} \left\Vert \mathbf {g}_t\right\Vert ^2_\infty \) 
